Certified Spectral Data for the GKW Operator

This page contains rigorously certified spectral data for the classical Gauss-Kuzmin-Wirsing transfer operator (s=1).

Certification timestamp: 2026-01-27T16:03:58.493

Certification Parameters

Certified Eigenvalues

The following eigenvalues are rigorously certified using resolvent bounds and the finite-to-infinite dimensional lift.

Notable Eigenvalues

• λ₁ ≈ 1: The Perron-Frobenius eigenvalue (invariant measure)
• λ₂ ≈ -0.3037: The Wirsing constant, determining the rate of convergence to the Gauss measure

Projection of Constant Function onto Eigenspaces

For the spectral expansion $L^n \cdot 1 = \sum_i \lambda_i^n \Pi_i(1)$, we compute the projection of the constant function 1 onto each eigenspace.

Eigenvector Coefficients

The eigenvectors are represented in the monomial basis $\{(x-1)^k\}_{k=0}^K$. Below are the first 10 coefficients of each certified eigenvector.

Eigenvector v_1 (λ = 1.0000000000)

v[0] = -8.660254037844e-01
v[1] = +4.330127018922e-01
v[2] = -2.165063509461e-01
v[3] = +1.082531754731e-01
v[4] = -5.412658773653e-02
v[5] = +2.706329386826e-02
v[6] = -1.353164693413e-02
v[7] = +6.765823467066e-03
v[8] = -3.382911733533e-03
v[9] = +1.691455866766e-03

Eigenvector v_2 (λ = -0.3036630029)

v[0] = -5.705092065901e-01
v[1] = -4.162454996025e-01
v[2] = +5.119616743648e-01
v[3] = -3.849198625430e-01
v[4] = +2.461699714070e-01
v[5] = -1.450824897558e-01
v[6] = +8.141216120369e-02
v[7] = -4.423293969454e-02
v[8] = +2.350035792038e-02
v[9] = -1.228659435422e-02

Eigenvector v_3 (λ = 0.1008845093)

v[0] = -9.044734494144e-02
v[1] = -7.068660107070e-01
v[2] = +9.448985678839e-02
v[3] = +3.086775037608e-01
v[4] = -4.051254176803e-01
v[5] = +3.476166459552e-01
v[6] = -2.496457140479e-01
v[7] = +1.620217337586e-01
v[8] = -9.845091317023e-02
v[9] = +5.712525784795e-02

Eigenvector v_4 (λ = -0.0354961590)

v[0] = -2.966845130687e-02
v[1] = -2.792405426546e-01
v[2] = -5.685386018879e-01
v[3] = +4.482418832737e-01
v[4] = -3.340947457413e-02
v[5] = -2.516780089683e-01
v[6] = +3.461878684946e-01
v[7] = -3.202397766613e-01
v[8] = +2.475727934510e-01
v[9] = -1.719036358233e-01

Eigenvector v_5 (λ = 0.0128437904)

v[0] = +8.804304398616e-03
v[1] = +1.282114165502e-01
v[2] = +4.062017203797e-01
v[3] = +2.504618933287e-01
v[4] = -5.386456910676e-01
v[5] = +3.438578833344e-01
v[6] = -1.768788118901e-02
v[7] = -2.147637445035e-01
v[8] = +3.075368335291e-01
v[9] = -3.000561734630e-01

Eigenvector v_6 (λ = -0.0047177775)

v[0] = +2.928800530413e-03
v[1] = +5.271351148338e-02
v[2] = +2.575842127988e-01
v[3] = +3.773320990927e-01
v[4] = -8.733264362254e-02
v[5] = -3.795465917734e-01
v[6] = +4.868057627563e-01
v[7] = -2.892103825029e-01
v[8] = +1.475498888265e-02
v[9] = +1.873669136701e-01

Eigenfunction Plots

Eigenfunction plots are generated by scripts/generate_eigenfunction_plots.jl and saved to data/. See the certification results PDF in docs/ for rendered figures.

Mathematical Details

Resolvent Bridge Theorem

The certification uses the resolvent bridge: if $\varepsilon_K \cdot \|R_{A_K}(z)\| < 1$ on a circle around λ, then the infinite-dimensional operator L has exactly one eigenvalue inside that circle.

Truncation Error

The truncation error is bounded by:

\[\varepsilon_K = C_2 \cdot \left(\frac{2}{3}\right)^{K+1}\]

where C₂ ≈ 10.0581.

Spectral Expansion

For the non-normal GKW operator:

\[L^n \cdot 1 = \sum_{i=1}^{m} \lambda_i^n \Pi_i(1) + O(|\lambda_{m+1}|^n)\]

The rate of convergence to the Gauss measure is determined by |λ₂/λ₁| ≈ 0.303663.